closed even once for diatoms. Curiously, in the research with Jacobson, we invoked the bilateral symmetry of the organism we were simulating, which both reduced computer time and made the simulation match reality much better. A similar two axis symmetry was placed on a pennate diatom simulation [2.12]. However, in a real organism there is no global “force” creating its symmetry. Symmetry is a consequence of morphogenesis, not a cause, and deviations from symmetry or its perfection are hints to the underlying processes.
2.1.4 Diatoms and Uncanny Symmetry
Microorganisms are generally not well studied in terms of quantified symmetry changes over time. Diatoms have distinct amorphous silica frustules that exhibit a variety of geometric shapes and surfaces that lend themselves to analyses of symmetries. Diatoms are pigmented protists that are considered to be a monophyletic phylum. Morphogenesis is a topic of great interest not only to phycologists but also to nanotechnologists [2.42, 2.44, 2.51, 2.85]. In girdle view, diatoms are asymmetrical because of the parent-daughter cell division that occurs within the previously formed cell. In valve view, shape and surface are both open to symmetry considerations.
We have proposed that diatoms possess “uncanny symmetry” [2.134]. By uncanny symmetry is meant that (some) diatoms in valve view (may) exhibit near perfect symmetry. It was shown that by subtracting the rotated image of a given diatom from its original image, an almost completely blacked out image would result, indicating near perfect matching of image pixels, suggesting near perfect symmetry [2.134]. Such results were obtained for Aulacodiscus oregonus and Triceratium formosum var. quinquelobata (Plate 7, Figures 1–4 in [2.134]). “The apparently high degree of perfection of this noncrystalline precipitate deserves quantification, which may prove comparable with the (small) degree of imperfection of crystalline snowflakes [2.83]” [2.134].
Centric diatoms exhibit rotational symmetry in shape and surface, but other symmetries such as dihedral symmetry are present as well. A case in point is Auliscus (Plate 5, Figure 1 in [2.134]). From dihedral symmetry, reflective symmetry of this diatom may be characterized as a 180° rotational symmetry, thereby identifying rotational symmetry as a starting point in the measurement of uncanny symmetry. That multiple symmetries occur simultaneously in centric diatom may be recovered by an uncanny symmetry measurement.
As suggested in [2.134], we propose to quantify centric diatom symmetry generally and uncanny symmetry specifically using concepts from information theory and image processing methods. Because symmetry involves the acquisition of information regarding the “balance” of an organism such as a diatom, information theory is used to develop a measure of centric diatom valve uncanny symmetry.
The incongruity of diatom morphogenesis and uncanny symmetry can be summarized in the question, why should diatoms possess uncanny symmetry? Symmetry is about pattern, and morphogenesis is about process. Why should this specific process produce a near perfect valve pattern and shape in diatoms? We propose to address this question by quantifying uncanny symmetry, quantifying the relation of uncanny symmetry to stability, and analyzing these results to determine the implications that symmetry via stability has on the diatom morphogenetic process.
2.1.5 Purpose of This Study
This study was conducted to measure rotational symmetry in centric diatoms and relate changes in symmetry to instability in a diatom morphogenetic dynamical system. We will demonstrate that symmetry is measurable explicitly as a deterministic quantity, and instability may be parsed to quantify deterministic and non-deterministic behavior in a morphogenetic dynamical system. Information contained in valve morphology is measurable via entropy, and this information is related to symmetry. Uncanny symmetry may be measured at multiple scales, and entropy as the measurement of this information is related to morphogenesis. Quantified centric diatom uncanny symmetry will be used to characterize behavior of symmetry changes for a variety of taxa. The relationship of uncanny symmetry to stability changes during centric diatom valve formation will be quantified using concepts from dynamical systems analysis. From here on out, we use the term symmetry to mean uncanny symmetry.
2.2 Methods
2.2.1 Centric Diatom Images Used for Analysis
Taxa were selected from a catalog of scanning electron micrographs (SEMs) in the library of one of the authors (M.A.T.). SEMs of centric diatom genera represented by selected species were used to quantify and analyze uncanny symmetry of diatom valve shape. Taxa selected for use represented circular, eccentric, and polygonal valve shapes that span the range of shapes of centric diatoms. A number of criteria were used to select images for analysis. Images were taken so that the view was determined to be perpendicular to the valve, where the valve appeared to be flat or at 0°. The whole valve had to be unbroken or in a state where breakage was qualitatively deemed to not impair the ability to discern the surface and/or shape boundary. The entire valve outline was unobstructed by debris, or if debris was present, it was qualitatively deemed to not detract from discerning the surface and/or shape boundary. Individuals were chosen to represent closely related species, and multiple species were selected from each genus when feasible.
Taxa whose images were chosen for study are given in Table 2.1. These 15 centric diatom genera represent a wide variety of genera, species, shapes, and valve patterns found in centric diatoms. Within each genus, species exhibit distinct features that are associated with the silica deposition process as evidenced by the morphology on the valve face. Morphological valve features are a record of silica deposition during valve formation and part of the morphogenetic process. The shape and position of valve features are indicators of symmetry of the valve surface.
Table 2.1 List of species considered and thumbnail, masked image examples for all of the individual diatom taxa we analyzed, all scaled to the same apparent diameter, along with individual number of rotations. Actual diameters vary with the life cycle. All thumbnails are external valves. All SEMs by Mary Ann Tiffany.
| Genus species | Thumbnail images | Rotations |
|---|---|---|
| 1. Actinoptychus | ||
| 1.1. Actinoptychus senarius |
|
3 |
| 1.2. Actinoptychus splendens |
|
14 |
| 2. Amphitetras antediluviana |
|
4 |
| 3. Arachnoidiscus | ||
| 3.1. Arachnoidiscus ehrenbergii |
|
14-31 |
| 3.2. Arachnoidiscus ornatus |
|
17-24
|